In the Armenian Church there are two days when he is commemorated. In 2009, they are January 24, where he is commemorated along with St. Athanasius and St. Gregory the Theologian, and October 24, where he is one of 12 theologians being commemorated. The dates will change slightly from year to year:

The rule for the first date may be something like "The third Saturday after Nativity (Jan. 6,)" and the rule for the second date may be something like "The Saturday before the Discovery of the Holy Cross," or "The fourth Saturday after the Cross of Varak," or "The sixth Saturday after the Exaltation of the Holy Cross," or something like that. It's probably the last one. I've always wished there was a book available for lay persons which explained how each saint day was calculated. Maybe some day a book like that will be available.

I've heard of a rule that says saints are commemorated on the anniversary of their death. However, I think there are other factors as well. For example, the first date above when St. Cyril is commemorated is also the date when two other theologians are celebrated. The second date has 12 theologians, including him. They can't all have died on the same date. So I am assuming that there is something else about those two times of the year that make it fitting to remember the prominent theologians of the Church.

I have a copy of a church calendar produced by Holy Virgin Mary Coptic Orthodox Church in Los Angeles (www.theotokos.org). They have an index in the back, indicating that the Coptic feast day for Cyril is July 10th. Can Coptic members here confirm this? Also, could someone explain why the feast day of Cyril would be on the July 10th, i.e. what is the significance of that date?